What Are They?
Damocloids are the inactive nuclei of Halley Family Comets.

How Do We Know That Damocloids Are Not Just Weird Asteroids?
On a case-by-case basis, we can often never be sure. However, we can be sure that the Damocloids as a group are related to the comets. This is because
a) some objects identified as Damocloids by the above criterion have subsequently developed comae: they are comets without a doubt
b) the distribution of inclinations of Damocloids is indistinguishable from that of the Halley Family Comets, even though the Tisserand selection criterion does not require that this be the case.
Figure
c) about 25% of the Damocloids possess retrograde orbits, unlike any other asteroids

What Do We Know About the Damocloids?
Just a little.

The albedos of four Damocloids have been measured and they are consistent in being amongst the lowest recorded for any Solar System objects (albedos are
a few %). The Damocloids are extremely dark.

The surfaces are redder than sunlight but the
ultrared matter that is found on many Kuiper Belt Objects and Centaurs is not present on the Damocloids. [
color-color figure from Jewitt 2005.]

The known examples are of modest size, with a median radius of 8 km.

What Would We Like to Know About the Damocloids?

How many of them are there? What is the ratio of the number of Damocloids to the number of HFCs? The answer to this question will tell us about the way in which HFCs die.

In what ways do the Damocloids differ, physically or compositionally, from the Halley Family Comets? In order to be inactive, they must possess devolatilized
("mantled") surfaces.
Are they different in other ways, perhaps as a result of repeated heating by the Sun when near perihelion?

How do the physical and compositional natures of the Damocloids compare with the nuclei of Jupiter Family Comets? Modification of the JFCs is profound,
including loss of the ultrared matter, but the Damocloid orbits are more extreme and these bodies appear generally less heated than the JFCs. Is there any
measurable effect on their properties?